This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

It depends on how it was installed. Update your question with the way the application was installed.
–
Mitch♦May 12 '13 at 8:46

Once you uninstall a program, the icon in the launcher will be removed. But if you only want to remove the icon from the launcher, and not removing the program, right-click over the icon and you will see the option to remove it.
–
DrKenobiMay 12 '13 at 13:33

Looks like you've found manuals relevant for old versions of Ubuntu which had System menu on top panel. It contained «Synaptic Package Manager» which is a great tool for installing and managing software. Currently, Ubuntu provides «Software Center» instead, but you still can use it to install Synaptic and then use Synaptic to install anything else.

I suppose, Ubuntu One can be removed (it is ubuntuone-client package in Synaptic), but be careful not to remove something important together with it. Synaptic always shows a warning when removing a package requires removing other packages, read it carefully in order to make sure that there is nothing important there. If there is, it's better not to remove Ubuntu One.

What is this "system"? System sounds important.

In Linux, most software is installed not only for current user but system-wide, so installing, removing and updating software requires administration privileges (also known as root privileges). That's why package management is considered a «system» operation and usually placed into «System» menu.

Anyway, if you just want to remove an icon from Unity left panel, right-click on it and remove. :)

I'll assume you're using unity, if so this video might help, or this tutorial. Also, just search the dash for Ubuntu one and Ubuntu music to open them without the launcher, like this tutorial says, under finding programs.